MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmtrsn Structured version   Visualization version   GIF version

Theorem pmtrsn 19223
Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)
Assertion
Ref Expression
pmtrsn (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn
Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5376 . . 3 {𝐴} ∈ V
2 eqid 2736 . . . 4 (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
32pmtrfval 19154 . . 3 ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
41, 3ax-mp 5 . 2 (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
5 eqid 2736 . . . . 5 (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
65dmmpt 6178 . . . 4 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7 2on0 8383 . . . . . . . . 9 2o ≠ ∅
8 ensymb 8863 . . . . . . . . . 10 (∅ ≈ 2o ↔ 2o ≈ ∅)
9 en0 8878 . . . . . . . . . 10 (2o ≈ ∅ ↔ 2o = ∅)
108, 9bitri 274 . . . . . . . . 9 (∅ ≈ 2o ↔ 2o = ∅)
117, 10nemtbir 3037 . . . . . . . 8 ¬ ∅ ≈ 2o
12 snnen2o 9102 . . . . . . . 8 ¬ {𝐴} ≈ 2o
13 0ex 5251 . . . . . . . . 9 ∅ ∈ V
14 breq1 5095 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
1514notbid 317 . . . . . . . . 9 (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16 breq1 5095 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
1716notbid 317 . . . . . . . . 9 (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
1813, 1, 15, 17ralpr 4648 . . . . . . . 8 (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
1911, 12, 18mpbir2an 708 . . . . . . 7 𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20 pwsn 4844 . . . . . . . 8 𝒫 {𝐴} = {∅, {𝐴}}
2120raleqi 3307 . . . . . . 7 (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
2219, 21mpbir 230 . . . . . 6 𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23 rabeq0 4331 . . . . . 6 ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
2422, 23mpbir 230 . . . . 5 {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
2524rabeqi 3416 . . . 4 {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
26 rab0 4329 . . . 4 {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
276, 25, 263eqtri 2768 . . 3 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
28 mptrel 5767 . . . 4 Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
29 reldm0 5869 . . . 4 (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
3028, 29ax-mp 5 . . 3 ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
3127, 30mpbir 230 . 2 (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
324, 31eqtri 2764 1 (pmTrsp‘{𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1540  wcel 2105  wral 3061  {crab 3403  Vcvv 3441  cdif 3895  c0 4269  ifcif 4473  𝒫 cpw 4547  {csn 4573  {cpr 4575   cuni 4852   class class class wbr 5092  cmpt 5175  dom cdm 5620  Rel wrel 5625  cfv 6479  2oc2o 8361  cen 8801  pmTrspcpmtr 19145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-1o 8367  df-2o 8368  df-er 8569  df-en 8805  df-pmtr 19146
This theorem is referenced by:  psgnsn  19224
  Copyright terms: Public domain W3C validator